Non-stationary analysis of a rotating shaft with geometrical nonlinearity during passage through critical speeds

In this paper, analysis of a rotating shaft with stretching nonlinearity during passage through critical speeds is investigated. In the model, the rotary inertia and gyroscopic effects are included, but shear deformation is neglected. The nonlinearity is due to large deflection of the shaft. First, nonlinear equations of motion governing the flexural–flexural–extensional vibrations of the rotating shaft with non-constant spin are derived by the Hamilton principle. Then, the equations are simplified using stretching assumption. To analyze the non-stationary vibration of the rotating shaft, the asymptotic method is applied to the equations expressed in normal coordinates. Two analytical expressions, as function of system parameters that describe the amplitude and phase of motion during passage through critical speeds are derived. The effects of angular acceleration, stretching nonlinearity, eccentricity and external damping on maximum amplitude of the shaft are investigated. It is shown that the nonlinearity has important effect on maximum amplitude when the rotating shaft passing through critical speeds, especially in low angular acceleration. To validate the results of the perturbation method, numerical simulation is applied.     

The effects of lateral–torsional coupling on the nonlinear dynamic behavior of a rotating continuous flexible shaft–disk system with rub–impact

This study investigates the lateral–torsional coupling effects on the nonlinear dynamic behavior of a rotating flexible shaft–disk system. The system is modeled as a continuous shaft with a rigid disk in its mid span. Coriolis and centrifugal effects due to shaft flexibility are also included. The partial differential equations of motion are extracted under the Rayleigh beam theory. The assumed mode method is used to discretize partial differential equations and the resulting equations are solved via numerical methods. The analytical methods used in this work include time series, phase plane portrait, power spectrum, Poincaré map, bifurcation diagrams, and Lyapunov exponents. The main objective of the present study is to investigate the torsional coupling effects on the chaotic vibration behavior of a system. Periodic, sub-harmonic, quasi-periodic, and chaotic states can be observed for cases with and without torsional effects. As demonstrated, inclusion of the torsional–lateral coupling effects can primarily change the speed ratios at which rub–impact occurs. Also, substantial differences are shown to exist in the nonlinear dynamic behavior of the system in the two cases.     

Vibrations and stability of axially traveling laminated beams

In this paper, vibrations and stability of an axially traveling laminated composite beam are investigated analytically via the method of multiple scales. Based on classical laminated beam theory, the governing equations of motion for a time-variant axial speed are obtained using Newton’s second law of motion and constitutive relations. The method of multiple scales, an approximate analytical method, is applied directly to the gyroscopic governing equations of motion and complex eigenfunctions and natural frequencies of the system are obtained. The stability boundaries of the system near resonance are determined via the Routh-Hurwitz criterion. Finally, a parametric study is conducted which considers the effects of laminate type and configuration as well as the mean speed and amplitude of speed fluctuations on the vibration response, natural frequencies and stability boundaries of the system.     

Nonlinear vibrations and stability of parametrically exited systems with cubic nonlinearities and internal boundary conditions: A general solution procedure

A general form of an analytical solution algorithm for the nonlinear vibrations and stability of parametrically excited continuous systems with intermediate concentrated elements is developed in this paper. The method of multiple timescales is applied directly to the equations of the motion which are in the form of a set of nonlinear partial differential equations with nonlinear coupled terms. This yields approximate analytical expressions for the response amplitude and stability of the system. Moreover, the solution to a sample problem is obtained using the general algorithm, thus proving its effectiveness and validity.     

Vibration of axially moving hyperelastic beam with finite deformation

In this paper, we study the vibration of an axially moving hyperelastic beam under simply supported condition. The kinematic of the axially moving beam have been described by Eulerian-Lagrangian formulation. In continuum mechanics frame, the finite deformation formula and a higher order shear deformation beam theory are applied to describe the deformation of the axially moving hyperelastic beam. In these formulas the material parameter, shear deformation and the geometric non-linearity have been taken into account. Through the Hamilton principle, the governing equations of nonlinear vibration are obtained, where the transverse vibration is coupled with the longitudinal vibration. When the velocity is a constant, the critical speed and natural frequencies are determined by solving the corresponding linear equations. Meantime, effects of the geometrical and material parameters on the critical speed and natural frequencies have been investigated. Comparisons among the critical velocities of the hyperelastic and Euler linear beam are also made. The results show that the critical velocity of hyperelastic beam is larger than that of linear Euler–Bernoulli beam. For the natural frequencies, we have the same conclusions. Lastly, by the multiple scales method, the leading order analytical solutions of the equilibrium state of axially moving hyperelastic beam in the supercritical regime are obtained. Furthermore the amplitudes of analytical solutions of the hyperelastic beam have been compared with that of linear Euler–Bernoulli beam. The effects of the material and geometrical parameters on the asymptotic solutions and the amplitude has been analyzed.     

三维弹性固体中冲击波传输方程的Lagrange描述

在Lagrange坐标中导出了三维非线性弹性固体中冲击波幅度在任意传播方向上的传输方程.导出的方程说明,冲击波的幅度在任意传播方向上随时间的变化率依赖于(i)冲击波阵面紧后方介质运动的一个未知量;(ii)冲击波阵面的两个主曲率;(iii)冲击波法向波速在阵面内的表面梯度;(iv)和冲击波前方介质运动有关的非齐次项,当前方介质处于均匀运动状态时此项为零.文中指出了适当选择传播矢量以简化传输方程的几种方法.我们还得到了一组与介质本构方程无关的、联系冲击波各跳跃量变化率的普适关系.     

Two-dimensional nonlinear dynamics of an axially moving viscoelastic beam with time-dependent axial speed

In the present study, the coupled nonlinear dynamics of an axially moving viscoelastic beam with time-dependent axial speed is investigated employing a numerical technique. The equations of motion for both the transverse and longitudinal motions are obtained using Newton’s second law of motion and the constitutive relations. A two-parameter rheological model of the Kelvin–Voigt energy dissipation mechanism is employed in the modelling of the viscoelastic beam material, in which the material time derivative is used in the viscoelastic constitutive relation. The Galerkin method is then applied to the coupled nonlinear equations, which are in the form of partial differential equations, resulting in a set of nonlinear ordinary differential equations (ODEs) with time-dependent coefficients due to the axial acceleration. A change of variables is then introduced to this set of ODEs to transform them into a set of first-order ordinary differential equations. A variable step-size modified Rosenbrock method is used to conduct direct time integration upon this new set of first-order nonlinear ODEs. The mean axial speed and the amplitude of the speed variations, which are taken as bifurcation parameters, are varied, resulting in the bifurcation diagrams of Poincaré maps of the system. The dynamical characteristics of the system are examined more precisely via plotting time histories, phase-plane portraits, Poincaré sections, and fast Fourier transforms (FFTs).     

Modélisation des coques non linéairement élastiques faiblement courbée en coordonnées curvilignes

The method of asymptotic expansions with the thickness as the small parameter is applied to the general three-dimensional equations for the equilibrium of a nonlinearly elastic shell. The problem is written in a weak form in curvilinear coordinates with the displacement as unknown. We show that the leading term of the asymptotic expansion can be identified with the solution of two-dimensional nonlinear shallow shell equations in curvilinear coordinates. In addition, we give an existence theorem and a regularity result for the two-dimensional nonlinear problem.     

Asymptotic periodicity of nonlinear discrete Volterra equations and applications

Sufficient conditions for the asymptotic periodicity of solutions of nonlinear discrete Volterra equations of Hammerstein type are obtained. Such results are applied to analyze the property of a class of numerical methods to preserve the asymptotic periodicity of the analytical solution of Volterra integral equations.     

Motion of an object-parachute system with allowance for canopy pulsations

The dynamics of an object-parachute system are examined with allowance for canopy pulsations. The law for the variation of the parachute's resistance is assumed to be harmonic. By means of the asymptotic method of nonlinear mechanics the equations for perturbed motion of the system with variable coefficients were investigated. Conditions were obtained for which the system has a parametric resonance. It is shown that in this case the amplitude of oscillations by the object-parachute system increases according to an exponential law.Translated from Dinamicheskie Sistemy, No. 5, pp. 67–73, 1986.     

液固耦合系统中液体的有限幅晃动力及晃动力矩

研究弹簧-质量系统与圆柱贮箱类液体有限幅晃动系统间的非线性耦合动力学问题。在建立了六自由度非线性耦合动力学模型的基础上,导出了液体有限幅晃动力和力矩解析表达式。指出在终了构形上积分及压力表达式中的非线性项是有限幅晃动作用力、作用力矩非线性的根源。x、y方向结果之间良好的对称性在很大程度上证明了结果的正确性。通过耦合机理分析可知,这样的理论结果应具有较大的普适性。数值仿真结果与有关实验结果进行了对比。分析认为,在终了构形上求晃动力、晃动力矩较为合理。舍去的高维模态基底及高阶非线性项以及液体晃动阻尼的复杂性是导致偏差的重要原因。     

稀相气固悬浮体越过沿平面匀速运动激波后的流动结构*

本文给出气固悬浮体中激波感生边界层的渐近数值分析,其中计及了作用于固体粒子的Saf-fman升力.研究结果表明粒子横越边界层的迁移导致了粒子轨道的交叉,因此对目前通用的含灰气体模型应做相应的修正.本文利用匹配渐近展开方法得到了匀速运动激波后方的两相侧壁边界层方程,详细描述了在Lagrange坐标下计算颗粒相流动参数的方法,并给出了粒子浓度很低情况下的数值结果.     

扁球壳和扁锥壳的轴对称非线性热弹耦合振动

假设温度场与应变场相互耦合,研究了旋转扁薄球壳和锥壳的轴对称非线性热弹振动问题．基于von Krmn理论和热弹性理论,导出了本问题的全部控制方程及其简化形式．应用Galerkin技术进行时空变量分离后,得到了一个关于时间的非线性常微分方程组．根据方程的特点,分别用多尺度法和正则摄动法求得了壳体振动的频率与振幅间特征关系和振幅衰减规律的一次近似解析解,并讨论了壳体几何参数、热弹耦合参数以及边界条件等因素对其非线性热弹耦合振动特性的影响．     

Lame equation and asymptotic higher-order periodic solutions to nonlinear evolution equations

Based on an auxiliary Lame equation and the perturbation method, a direct method is proposed to construct asymptotic higher-order periodic solutions to some nonlinear evolution equations. It is shown that some asymptotic higher-order periodic solutions to some nonlinear evolution equations in terms of Jacobi elliptic functions are explicitly obtained with the aid of symbolic computation.     

Superposition principle and composite solutions to coupled nonlinear Schrödinger equations

We show that the superposition principle applies to coupled nonlinear Schrödinger equations with cubic nonlinearity where exact solutions may be obtained as a linear combination of other exact solutions. This is possible due to the cancelation of cross terms in the nonlinear coupling. First, we show that a composite solution, which is a linear combination of the two components of a seed solution, is another solution to the same coupled nonlinear Schrödinger equation. Then, we show that a linear combination of two composite solutions is also a solution to the same equation. With emphasis on the case of Manakov system of two-coupled nonlinear Schrödinger equations, the superposition is shown to be equivalent to a rotation operator in a two-dimensional function space with components of the seed solution being its coordinates. Repeated application of the rotation operator, starting with a specific seed solution, generates a series of composite solutions, which may be represented by a generalized solution that defines a family of composite solutions. Applying the rotation operator to almost all known exact seed solutions of the Manakov system, we obtain for each seed solution the corresponding family of composite solutions. Composite solutions turn out, in general, to possess interesting features that do not exist in the seed solution. Using symmetry reductions, we show that the method applies also to systems of N-coupled nonlinear Schrödinger equations. Specific examples for the three-coupled nonlinear Schrödinger equation are given.     

Waves of polarized light in nonlinear birefringent fibre

A propagation of a short optical pulse in nonlinear birefringent fibre is described by a system of two coupled Schrödinger equations. By means of variational Anderson method this system reduces to the system of ordinary differential equations for spatial evolution of pulse parameters. In two ultimate cases the analytical solutions of the equations are managed to be found. It is shown that at some critical power of the input pulse W_c the regime of propagation changes. For the power exceeding W_c the radiation concentrates in one channel. The numerical investigation of the intermediate cases was done when by the variation of the input pulse power one can achieve the comparable effectiveness of the competing processes of dispersion broadening and nonlinear pulse compression. The numerical simulations show that in the range of critical values of the nonlinear coupling coefficient the transition takes place to the chaotic phase and amplitude behavior of the coupled waves of different polarizations. The research is important to understand the processes of ultra short digital pulses propagation in optical fibre links.     

具变时滞的非线性退化微分系统的渐近稳定性判据

In this paper, the asymptotic stability for singular differential nonlinear systems with multiple time-varying delays is considered. The V-functional method for general singular differential delay system is investigated. The asymptotic stability criteria for singular differential nonlinear systems with multiple time-varying delays are derived based on V-functional method and some analytical techniques, which are described as matrix equations or matrix inequalities. The results obtained are computationally flexible and efficient.     

Effects of Cone Angles on Nonlinear Vibration Responses of Functionally Graded Shells北大核心CSCD

The nonlinear vibration responses of functionally graded materials (FGMs) shells with different cone angles under external loads were studied. Firstly, the Voigt model was employed to describe the physical properties along the thickness direction of FGMs conical shells. Then, the motion equations were derived based on the 1st-order shear deformation theory, the von Kármán geometric nonlinearity and Hamilton’s principle. Next, the Galerkin method was applied to discretize the motion equations and the governing equations were simplified into a 1DOF nonlinear vibration differential equation under Volmir’s assumption. Finally, the nonlinear motion equations were solved with the harmonic balance method and the Runge-Kutta method, and the amplitude frequency response characteristic curves of the FGMs conical shells were obtained. The effects of different material distribution functions and different ceramic volume fraction exponents on the amplitude frequency response curves of conical shells were discussed. The bifurcation diagrams of conical shells with different cone angles, as well as time process diagrams and phase diagrams for different excitation amplitudes, were described. The motion characteristics were characterized by Poincaré maps. The results show that, the FGMs conical shells present the nonlinear characteristics of hardening springs. The chaotic motions of the FGMs conical shells are restrained and not prone to motion instability with the increase of the cone angle. The FGMs conical shell present a process from the periodic motion to the multi-periodic motion and then to chaos with the increase of the excitation amplitude. © 2022 Editorial Office of Applied Mathematics and Mechanics. All rights reserved.     

Analysis of the effects of a pulsed electromagnetic field on the dynamic response of electrically conductive composites

The effects of pulsed electromagnetic fields on the dynamic mechanical response of electrically conductive anisotropic plates are studied. The analysis is based on the simultaneous solving of the system of nonlinear partial differential equations that include equations of motion and Maxwell’s equations. Physics-based hypotheses for electro-magneto-mechanical coupling in anisotropic composite plates and dimension reduction solution procedures for the nonlinear system of the governing equations are presented. A numerical solution procedure for the resulting two-dimensional nonlinear system of the governing equations has been developed and consists of the sequential application of time and spatial integration and quasilinearization. The developed methodology is applied to the problem of the dynamic response of a long current-carrying unidirectional carbon fiber polymer matrix composite plate subjected to transverse impact load and in-plane pulsed electromagnetic load. The interacting effects of the pulsed electric current, external magnetic field, and mechanical load are studied.